Superlinear elliptic inequalities on manifolds (Q2295694)

The article are concerned with the existence and nonexistence of solutions for the inequality \(-\Delta u\geq \sigma u^q\) on a connected complete non-compact manifold \(M\). Here \(\sigma\) is a nonnegative Radon measure on \(M\) and \(q>1\) is a real number. The authors deduce necessary and suficient conditions in terms of the Green function that corresponds to the Laplace operator. These conditions are in explicit form when \(M\) has nonnegative Ricci curvature on \(M\) or when the Green function satisfies the \(3G\)-inequality.